# Sequences and Series: Modelling

## Understanding Modelling of Sequences and Series

• Modelling involves using a mathematical structure to represent real-world phenomena and then to predict or explain the phenomena.
• In sequences and series, the modelling process may involve setting up a sequence to describe a situation, or fitting a sequence to data.
• A series is the summed terms of a sequence, whereas the sequence is simply the list of numbers.

## Arithmetic Sequences and Series

• An arithmetic sequence is a sequence of numbers in which the difference of between any two successive members is constant.
• The nth term of an arithmetic sequence can be given by a + (n-1)d, where a is the first term and d is the common difference.
• The sum of an arithmetic series, `S_n`, of n terms can be calculated using the formula S_n = n/2(2a + (n - 1)d).

## Geometric Sequences and Series

• A geometric sequence is a sequence of numbers in which the ratio of any two successive members is constant. This constant is often called the common ratio.
• The nth term of a geometric sequence can be given by ar^(n-1), where a is the first term and r is the common ratio.
•  The sum of a geometric series, `S_n`, of n terms can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where r < 1.

## Using Sequences and Series in Modelling

• In modelling physical or biological phenomena, arithmetic and geometric sequences and series are often used to represent or approximate the phenomena.
• The choice between arithmetic and geometric sequences and series often depends on the nature of the phenomena, such as whether the phenomena exhibit constant change or exponential change.

## The Limit of a Series

• The limit of a series is a value that the series approaches as the number of terms goes to infinity.
•  For a geometric series where r < 1, the series has a limit, given by a / (1 - r).

## Fibonacci and Other Sequences

• There are many sequences other than arithmetic and geometric sequences, such as the Fibonacci sequence, which can also be used in modelling.
• Fibonacci sequence is defined by the recurrence relation f(n) = f(n - 1) + f(n - 2) with initial conditions f(0) = 0 and f(1) = 1.
• In a similar way to arithmetic and geometric sequences, the properties of these sequences can be analysed, and the sequences can be used to model real-world phenomena.